class: center, middle, inverse, title-slide .title[ # Cálculo Aplicado à Finanças ] .subtitle[ ## Parte I ] .author[ ### Jailson Duarte ] .institute[ ### GFIC/UFPB ] .date[ ### 12/04/2025 (Updated: 21 de abril de 2025) ] --- # 1. Conceito de Função -- - **Uso de funções matemática na análise de fenômenos econômicos:** -- - Cálculo de receitas e lucros - Avaliação de investimentos (valor presente e futuro) - Análise de custos (fixos e variáveis) - Depreciação de ativos - Modelagem de oferta e demanda - Análise de risco e retorno de ativos financeiros - Decisões de financiamento -- - **Função**: é uma regra que associa a cada elemento de um conjunto **A** um único elemento de um conjunto **B**: y = f(x). -- - A cada valor da **x** (variável independente) está associado um único valor de **y** (variável dependente). --- # Exemplo de Função -- $$ D(t) = \frac{V_i - V_r}{V_u} \cdot t $$ - `\({V_i}\)` = Valor Inicial; - `\({V_r}\)` = Valor Residual; - `\({V_u}\)` = Vida Útil; <img src="data:image/png;base64,#index_files/figure-html/unnamed-chunk-1-1.png" style="display: block; margin: auto;" /> --- # Tipos de Função -- ### 📈 Função Crescente -- - f(x) é **crescente**: em um intervalo, ao aumentar o valor de x, o valor de f(x) também aumenta. `\(x_1 < x_2 \Rightarrow f(x_1) \leq f(x_2)\)`. **Exemplo**: f(x) = 2x + 1. -- ### 📉 Função Decrescente - f(x) é **decrescente**: em um intervalo, ao aumentar o valor de x, o valor de f(x) diminui. `\(x_1 < x_2 \Rightarrow f(x_1) \geq f(x_2)\)`. **Exemplo**: f(x) = -3x + 10. --- ## Curvas de Oferta e Demanda <img src="data:image/png;base64,#index_files/figure-html/unnamed-chunk-2-1.png" style="display: block; margin: auto;" /> --- # Tipos de Função -- ### 🔒 Função Limitada - Quando seus valores (a imagem) ficam dentro de um intervalo específico. `\(f(x)\)` é **limitada** em um intervalo `\(I \subseteq \mathbb{R}\)` se **existem dois números reais** `\(M\)` e `\(m\)` tais que `\(m \leq f(x) \leq M \quad \text{para todo } x \in I\)` -- .pull-left[ <img src="data:image/png;base64,#index_files/figure-html/unnamed-chunk-3-1.png" style="display: block; margin: auto;" /> ] .pull-right[ <span style="font-size: 0.7em;"> Opção de compra (call) com *payoff* limitado. O investidor não pode ganhar mais do que um certo valor máximo `\(C_{\text{max}}\)`. `\(f(S) = \min\left( \max(S - K, 0), C_{\text{max}} \right)\)`. Onde: `\(S\)`: preço do ativo no vencimento; `\(K\)`: preço de exercício (strike); `\(C_{\text{max}}\)`: *payoff* máximo; </span> ] --- # Tipos de Função -- ### 🔀 Função Composta -- - `\(f: B \rightarrow C\)` e `\(g: A \rightarrow B\)` duas funções. A **função composta** `\(f \circ g\)` é definida por `\((f \circ g)(x) = f(g(x))\)` para todo `\(x \in A\)`, tal que `\(g(x) \in B\)`. **Exemplo**: `\(g(x) = x^2\)` e `\(f(y) = sen y\)`, então `\((f \circ g)(x) = sen(x^2)\)`. <img src="data:image/png;base64,#index_files/figure-html/unnamed-chunk-4-1.png" style="display: block; margin: auto;" /> --- # 2 Modelos Lineares -- Uma **função linear**: `\(f(x) = a \cdot x + b\)`. - `\(a\)` é o **coeficiente angular** da reta, que determina a inclinação da linha. É a taxa de variação de `\(y\)` em relação a `\(x\)`. Ou seja, para cada unidade que `\(x\)` aumenta, `\(y\)` aumenta (ou diminui, dependendo do sinal de `\(a\)`) em `\(a\)` unidades~; - `\(b\)` é o **coeficiente linear** (ou **intercepto**), que é o valor de `\(f(x)\)` quando `\(x = 0\)`. Este valor é o ponto onde a reta cruza o eixo `\(y\)`. - **Exemplo**: função de custo total: `\(C(q) = C_f + C_v \cdot q\)`, com `\(C_f\)` = **custo fixo** (independente da produção); `\(C_v\)` = **custo variável unitário** e `\(q\)` = **quantidade produzida**. --- ## Função de Custo, Receita e Ponto de Equilíbrio -- <img src="data:image/png;base64,#index_files/figure-html/unnamed-chunk-5-1.png" style="display: block; margin: auto;" /> --- # Regressão Linear Simples -- - **Função linear**: relação matemática determinística entre duas variáveis. - **Regressão linear simples**: modelo estatístico usado para **estimar** a relação entre uma variável independente `\(x\)` e uma variável dependente `\(y\)`, com base em dados observados: `\(y = \alpha + \beta x + \varepsilon\)`. -- Onde: - `\(y\)`: variável dependente (ou variável resposta) - `\(x\)`: variável independente (ou variável explicativa) - `\(\alpha\)`: intercepto da reta de regressão - `\(\beta\)`: coeficiente angular (inclinação da reta) - `\(\varepsilon\)`: termo de erro aleatório, que representa a diferença entre o valor observado e o valor estimado --- ## Exemplo da Resta Estimada <img src="data:image/png;base64,#index_files/figure-html/unnamed-chunk-6-1.png" style="display: block; margin: auto;" /> -- <span style="font-size: 0.8em;"> A equação estimada da regressão linear simples é: `\(\hat{y} = \hat{\alpha} + \hat{\beta} x\)`. Com `\(\hat{y}\)`: valor estimado da variável dependente; `\(x\)`: variável independente (observada); `\(\hat{\alpha}\)`: estimativa do intercepto da reta de regressão; `\(\hat{\beta}\)`: estimativa do coeficiente angular (inclinação da reta). </span> --- # Função do 2º Grau -- Uma **função quadrática** (ou função polinomial do 2º grau) é dada por: `\(f(x) = a x^2 + b x + c\)` Onde: - `\(a\)`: coeficiente quadrático (determina a concavidade da parábola) - `\(b\)`: coeficiente linear (inclinação da reta tangente no ponto de origem) - `\(c\)`: coeficiente constante (valor de `\(f(x)\)` quando `\(x = 0\)`) --- ## Aplicação: Função de Custo com Efeito de Economia de Escala -- Considere uma situação onde há economia de escala ou ineficiências crescentes, fazendo com que o custo total aumente de forma **não-linear** à medida que a produção cresce. Uma função quadrática pode representar isso: `\(C(q) = a q^2 + b q + c\)` - `\(a > 0\)`: representa o efeito crescente de custo conforme aumenta a produção. - `\(b\)`: representa o custo variável linear. - `\(c\)`: é o custo fixo inicial. --- ## Gráfico: Custo Total Quadrático <img src="data:image/png;base64,#index_files/figure-html/unnamed-chunk-7-1.png" style="display: block; margin: auto;" /> --- ## Regressão Polinomial de Grau 2 -- A regressão polinomial de segundo grau é uma extensão da regressão linear, que permite modelar relações não-lineares entre as variáveis: `\(\hat{y} = \hat{\alpha} + \hat{\beta}_1 x + \hat{\beta}_2 x^2\)` Onde: `\(\hat{\alpha}\)`: intercepto estimado `\(\hat{\beta}_1\)`: coeficiente linear estimado `\(\hat{\beta}_2\)`: coeficiente quadrático estimado `\(x\)`: variável independente --- ## Exemplo <img src="data:image/png;base64,#index_files/figure-html/unnamed-chunk-8-1.png" style="display: block; margin: auto;" /> --- # Introdução ao Cálculo 1. Encontrar a reta tangente a uma curva em dado ponto da curva 2. Encontrar a área da região plana limitada por uma curva arbitrária -- <img src="data:image/png;base64,#index_files/figure-html/unnamed-chunk-9-1.png" style="display: block; margin: auto;" /> --- # Noção Intuitiva de Limites Vamos considerar a função: `\(g(t) = \frac{4(t^2 - 4)}{t - 2}\)` A função **não está definida** em `\(t = 2\)`. #### Cálculo do Limite `$$g(t) = \frac{4(t^2 - 4)}{t - 2} = \frac{4(t - 2)(t + 2)}{t - 2}$$` Para `\(t \neq 2\)`, a função pode ser simplificada: `\(g(t) = 4(t + 2)\)`. Logo, o limite quando `\(t \to 2\)` é: `$$\lim_{t \to 2} g(t) = 4(2 + 2) = 4 \cdot 4 = 16$$` --- ## Visualização Gráfica <img src="data:image/png;base64,#index_files/figure-html/unnamed-chunk-10-1.png" style="display: block; margin: auto;" /> --- # Limite de uma Função A função `\(f\)` tem **limite `\(L\)`** quando `\(x\)` se aproxima de `\(a\)`, o se denota por: $$ \lim_{x \to a} f(x) = L $$ se pudermos fazer o valor de `\(f(x)\)` **tão próximo de `\(L\)`** quanto quisermos, **bastando tomar `\(x\)` suficientemente próximo (mas não igual) a `\(a\)`**. --- ## Propriedades dos Limites Suponha que: `\(\lim_{x \to a} f(x) = L \quad \text{e} \quad \lim_{x \to a} g(x) = M\)` Então, valem as seguintes propriedades: 1. `\(\displaystyle{\lim_{x \to a}} [f(x)]^r = \displaystyle{[\lim_{x \to a}} f(x)]^r = L^r ,\quad \text{com } r \in \mathbb{R}\)` -- 2. `\(\displaystyle{\lim_{x \to a} c} \cdot f(x) = c \cdot \displaystyle{\lim_{x \to a}} f(x) = cL, \quad \text{com } c \in \mathbb{R}\)` -- 3. `\(\displaystyle{\lim_{x \to a}} [f(x) \pm g(x)] = \lim_{x \to a} f(x) \pm \lim_{x \to a} g(x) = L \pm M\)` -- 4. `\(\displaystyle{\lim_{x \to a}} [f(x) \cdot g(x)] = [\lim_{x \to a} f(x) ] \cdot [\lim_{x \to a} g(x)] = LM\)` -- 5. `\(\displaystyle{\lim_{x \to a}} \frac{f(x)}{g(x)} = \frac{\displaystyle{\lim_{x \to a}} f(x)}{\displaystyle{\lim_{x \to a} g(x)}} = \frac{L}{M}, \quad \text{com } M \neq 0\)` --- ## Exemplos Cálculo dos Limites -- 1. `\(\displaystyle{\lim_{x \to 1}}(5x^4 - 2) = \displaystyle{\lim_{x \to 1} 5x^4 } - \displaystyle{\lim_{x \to 1} 2}= 5(1)^4 - 2 = 3\)` -- 2. `\(\displaystyle{\lim_{x \to 3}} (x^2 \cdot \sqrt{x}) = \left( \lim_{x \to 3} x^2 \right) \cdot \left( \lim_{x \to 3} \sqrt{x} \right) = 3^2 \cdot \sqrt{3}= 9 \cdot \sqrt{3}\)` -- 3. `\(\displaystyle{\lim_{x \to 2}} \frac{x^2 + 2x}{x + 1} = \frac{\displaystyle{\lim_{x \to 2}} (x^2 + 2x)}{\displaystyle{\lim_{x \to 2}} (x + 1)}=\frac{2^2 + 2 \cdot 2 }{2 + 1}=\frac{4 + 4 }{2 + 1}=\frac{8}{3}\)` --- # Limites no Infinito -- Há situações em que queremos saber se f(x) tende a um único número quando `\(x\)` cresce além de qualquer limite. -- <img src="data:image/png;base64,#index_files/figure-html/unnamed-chunk-11-1.png" style="display: block; margin: auto;" /> --- # TEOREMA Para todo `\(n > 0\)`, temos: `\(\displaystyle{\lim_{x \to \infty}} \frac{1}{x^n} = 0 \quad \text{e} \quad \displaystyle{\lim_{x \to -\infty}} \frac{1}{x^n} = 0\)` Desde que `\(\frac{1}{x^n}\)` esteja definido. -- **Exemplo**: Vamos considerar a função `\(f(x) = \frac{1}{x^2}\)` Quando `\(x \to \infty\)` ou `\(x \to -\infty\)`, o valor de `\(f(x)\)` se aproxima de 0. Portanto, `\(\displaystyle{\lim_{x \to \infty}} \frac{1}{x^2} = 0 \quad \text{e} \quad \displaystyle{\lim_{x \to -\infty}} \frac{1}{x^2} = 0\)` --- # Funções Contínuas De modo informal, uma função é contínua em um ponto se seu gráfico naquele ponto não apresenta buracos, saltos ou quebras. <img src="data:image/png;base64,#index_files/figure-html/unnamed-chunk-12-1.png" width="1440" style="display: block; margin: auto;" /> -- Uma função `\(f(x)\)` é **contínua** no ponto `\(x = a\)` se as seguintes condições forem satisfeitas: **1.** `\(f(a)\)` está definida; **2.** `\(\displaystyle \lim_{x \to a} f(x)\)` existe; **3.** `\(\displaystyle \lim_{x \to a} f(x) = f(a)\)` --- # Conceito de Derivada -- Considere a função de produção total `\(Q\)` dependende de um insumo `\(x\)`: `\(Q = f(x)\)`. -- Aumentando o insumo de `\(x\)` para `\(x + \Delta x\)`, a produção muda de `\(f(x)\)` para `\(f(x + \Delta x)\)`. A **taxa de variação média** da produção nesse intervalo é: $$ \frac{f(x + \Delta x) - f(x)}{\Delta x} $$ -- Essa razão indica **o quanto, em média, a produção aumenta a cada unidade adicional do insumo** no intervalo `\(\Delta x\)`. --- ## Taxa de Variação Instantânea A **derivada** é definida como o limite da taxa de variação média quando o intervalo tende a zero: -- $$ f'(x) = \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x} $$ -- A derivada `\(f'(x)\)` representa o **produto marginal do insumo**, ou seja, quanto a produção aumenta ao se adicionar uma unidade extra do insumo, mantendo os demais fatores constantes. -- - Se `\(x\)` for o número de trabalhadores, então `\(f'(x)\)` é o **produto marginal do trabalho**. --- ## Exemplo com Δx tendendo a zero A derivada de `\(f(x) = x^2\)` é dada por `\(f'(x) = \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}=2\)`. -- <div id="ojmgiijwug" style="padding-left:0px;padding-right:0px;padding-top:10px;padding-bottom:10px;overflow-x:auto;overflow-y:auto;width:auto;height:auto;"> <style>#ojmgiijwug table { font-family: system-ui, 'Segoe UI', Roboto, Helvetica, Arial, sans-serif, 'Apple Color Emoji', 'Segoe UI Emoji', 'Segoe UI Symbol', 'Noto Color Emoji'; 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} </style> <table class="gt_table" data-quarto-disable-processing="false" data-quarto-bootstrap="false"> <thead> <tr class="gt_heading"> <td colspan="4" class="gt_heading gt_title gt_font_normal gt_bottom_border" style>Derivada de f(x) = x² com diferentes Valores de Δx</td> </tr> <tr class="gt_col_headings gt_spanner_row"> <th class="gt_center gt_columns_top_border gt_column_spanner_outer" rowspan="1" colspan="4" scope="colgroup" id="Resultados"> <span class="gt_column_spanner">Resultados</span> </th> </tr> <tr class="gt_col_headings"> <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" style="font-weight: bold;" scope="col" id="ValorX">ValorX</th> <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" style="font-weight: bold;" scope="col" id="DeltaX">DeltaX</th> <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" style="font-weight: bold;" scope="col" id="TxAprox">TxAprox</th> <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" style="font-weight: bold;" scope="col" id="TxReal">TxReal</th> </tr> </thead> <tbody class="gt_table_body"> <tr><td headers="ValorX" class="gt_row gt_center">1.00000000</td> <td headers="DeltaX" class="gt_row gt_center">1.00000000</td> <td headers="TxAprox" class="gt_row gt_center">3.00000000</td> <td headers="TxReal" class="gt_row gt_center">2.00000000</td></tr> <tr><td headers="ValorX" class="gt_row gt_center">1.00000000</td> <td headers="DeltaX" class="gt_row gt_center">0.10000000</td> <td headers="TxAprox" class="gt_row gt_center">2.10000000</td> <td headers="TxReal" class="gt_row gt_center">2.00000000</td></tr> <tr><td headers="ValorX" class="gt_row gt_center">1.00000000</td> <td headers="DeltaX" class="gt_row gt_center">0.01000000</td> <td headers="TxAprox" class="gt_row gt_center">2.01000000</td> <td headers="TxReal" class="gt_row gt_center">2.00000000</td></tr> <tr><td headers="ValorX" class="gt_row gt_center">1.00000000</td> <td headers="DeltaX" class="gt_row gt_center">0.00100000</td> <td headers="TxAprox" class="gt_row gt_center">2.00100000</td> <td headers="TxReal" class="gt_row gt_center">2.00000000</td></tr> <tr><td headers="ValorX" class="gt_row gt_center">1.00000000</td> <td headers="DeltaX" class="gt_row gt_center">0.00000010</td> <td headers="TxAprox" class="gt_row gt_center">2.00000010</td> <td headers="TxReal" class="gt_row gt_center">2.00000000</td></tr> </tbody> </table> </div> --- ## Interpretação Gráfica -- <img src="data:image/png;base64,#img/GrafDerivada.gif" style="display: block; margin: auto;" /> <div style="text-align: center; font-size: 80%;"> Fonte: <a href="https://www.youtube.com/watch?v=azcIK8kcN7Q">Youtube</a> </div> --- ## Derivada de f em x -- A derivada de uma função `\(f\)` é a função `\(f'\)` que fornece a declividade da reta tangente ao gráfico de `\(f\)` em qualquer ponto `\((x,f(x))\)` e também a taxa de variação de `\(f\)` em `\(x\)`. - Outras Notações para a Derivada -- - `\(D_xf(x)\)`: *Leia-se:* "d sub x de f de x" - `\(\frac{dy}{dx}\)`: *Leia-se:* "d y d x" - `\(y'\)`: *Leia-se:* "y linha" --- ## Regras de Derivação -- 📌 **Derivada de uma constante** Seja `\(c\)` uma constante, então: `\(\frac{d}{dx} (c) = 0\)`. 🔍 **Exemplo:** `\(\frac{d}{dx}(7) = 0\)` -- 📌 **Regra da potência** Para qualquer número real `\(n\)`: `\(\frac{d}{dx}(x^n) = n x^{n-1}\)`. 🔍 **Exemplo:** `\(\frac{d}{dx}(x^5) = 5x^4\)` --- ## Regras de Derivação -- 📌 **Derivada de uma constante multiplicada por uma função** Seja `\(c\)` uma constante, então: `\(\frac{d}{dx}[c \cdot f(x)] = c \cdot f'(x)\)`. 🔍 **Exemplo:** `\(\frac{d}{dx}(3x^2) = 3 \cdot \frac{d}{dx}(x^2) = 3 \cdot 2x = 6x\)` -- 📌 **Regra da soma (ou subtração)** `\(\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)\)`. 🔍 **Exemplo:** `\(\frac{d}{dx}(x^2 + 4x) = \frac{d}{dx}(x^2) + \frac{d}{dx}(4x) = 2x + 4\)` --- ## Regras de Derivação -- 📌 **Regra do produto** `\(\frac{d}{dx}[f(x) \cdot g(x)] = f'(x) \cdot g(x) + f(x) \cdot g'(x)\)` 🔍**Exemplo:** `\(\frac{d}{dx}(x^2 \cdot x^3) = 2x \cdot x^3 + x^2 \cdot 3x^2 = 2x^4 + 3x^4 = 5x^4\)` -- 📌 **Regra do quociente** `\(\frac{d}{dx} \left[\frac{f(x)}{g(x)}\right] = \frac{f'(x) \cdot g(x) - f(x) \cdot g'(x)}{[g(x)]^2}\)` 🔍**Exemplo:** `\(\frac{d}{dx} \left( \frac{x^2}{x^3} \right) = \frac{(2x) \cdot x^3 - x^2 \cdot 3x^2}{(x^3)^2} = \frac{2x^4 - 3x^4}{x^6} = \frac{-x^4}{x^6} = -\frac{1}{x^2}\)` --- ## Regras de Derivação -- 📌 **Regra da cadeia** Se `\(y = f(g(x))\)`, então: `\(\frac{dy}{dx} = f'(g(x)) \cdot g'(x)\)` De modo equivalente, se escrevermos `\(y = h(x) = g(u)\)`, onde `\(u = f(x)\)`, então `\(\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}\)` 🔍 **Exemplo:** `\(f(x) = (3x^2 + 2)^5\)` Aplicando a **regra da cadeia**: `$$\frac{d}{dx} \left[ (3x^2 + 2)^5 \right] = 5(3x^2 + 2)^4 \cdot (6x) = 30x(3x^2 + 2)^4$$` --- ## Regras de Derivação -- 📌 **Derivada da função exponencial** A derivada de `\(e^x\)` em relação a `\(x\)` é dada por: `\(\frac{d}{dx} e^x = e^x\)` 🔍 **Exemplo:** `\(\frac{d}{dx} e^{2x} = 2e^{2x}\)` -- 📌 **Derivada do logaritmo natural** A derivada do logaritmo natural `\(\ln(x)\)` é dada por: `\(\frac{d}{dx} \ln(x) = \frac{1}{x}\)` 🔍 **Exemplo:** `\(\frac{d}{dx} \ln(3x) = \frac{1}{3x} \cdot 3 = \frac{1}{x}\)` --- ## Regras de Derivação -- 📌 **Derivada da função exponencial com base `\(a\)`** Para funções exponenciais com base `\(a\)`, temos: `\(\frac{d}{dx} a^x = a^x \ln(a)\)` 🔍 **Exemplo:** `\(\frac{d}{dx} 2^x = 2^x \ln(2)\)` -- 📌 Derivada de uma função composta com a exponencial Se `\(y = e^{g(x)}\)`, a derivada é dada por: `\(\frac{d}{dx} e^{g(x)} = e^{g(x)} \cdot g'(x)\)` 🔍 **Exemplo:** `\(\frac{d}{dx} e^{x^2} = e^{x^2} \cdot 2x\)` --- # Derivadas de Ordem Superior -- - Se `\(f(x)\)` é diferenciável, então sua primeira derivada é `\(f'(x)\)`, e a derivada da derivada, ou seja, `\(f''(x)\)`, é a **segunda derivada**. De forma geral, a n-ésima derivada é `\(f^{(n)}(x)\)`. Primeira derivada: `\(\frac{dy}{dx}\)` Segunda derivada: `\(\frac{d^2y}{dx^2}\)` Terceira derivada: `\(\frac{d^3y}{dx^3}\)` Enésima derivada: `\(\frac{d^n y}{dx^n}\)` --- # Polinômio de Taylor -- Se uma função `\(f(x)\)` é infinitamente diferenciável em torno de um ponto `\(a\)`, o **Polinômio de Taylor** de ordem `\(n\)` para `\(f(x)\)` é dado por: `\(P_n(x) = f(a) + \frac{f'(a)}{1!}(x - a) + \frac{f''(a)}{2!}(x - a)^2 + \cdots + \frac{f^{(n)}(a)}{n!}(x - a)^n\)` -- Ou, de forma compacta: `\(P_n(x) = \sum_{k=0}^{n} \frac{f^{(k)}(a)}{k!} (x - a)^k\)` Esse polinômio aproxima a função `\(f(x)\)` perto do ponto `\(a\)` com base em suas derivadas sucessivas avaliadas em `\(a\)`. --- # Aproximação de `\(e^x\)` -- Vamos construir o polinômio de Taylor de ordem 4 para `\(f(x) = e^x\)`, centrado em `\(a = 0\)`. As derivadas são: `\(f(x) = e^x\)`; `\(f'(x) = e^x\)`; `\(f''(x) = e^x\)`; `\(f^{(k)}(x) = e^x\)` para todo `\(k\)`. Logo, `\(f^{(k)}(0) = 1\)` para todo `\(k\)`. -- **Exemplo**: Aproximação de `\(e^x\)` `$$P_4(x) = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} = 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \frac{x^4}{24}$$` Esse polinômio aproxima `\(e^x\)` para valores próximos de `\(x = 0\)`. --- # Juros Compostos -- **Juros compostos contínuos:** `\(M(t) = P \cdot e^{rt}\)`. Podemos usar o polinômio de Taylor para aproximar `\(e^{rt}\)` quando a taxa de juros é pequena. Aproximação de `\(e^{rt}\)` com Taylor de ordem 2 centrado em 0: `$$e^{rt} \approx 1 + rt + \frac{(rt)^2}{2}$$` Logo, o montante aproximado é: `$$M(t) \approx P \left(1 + rt + \frac{(rt)^2}{2} \right)$$` --- # 🧮 Cálculo em R - Parâmetros - Capital inicial: P = 1.000 - Taxa de juros (5% ao ano): r = 0.05 - Tempo (1 ano): t = 1. <div id="ufybehlmra" style="padding-left:0px;padding-right:0px;padding-top:10px;padding-bottom:10px;overflow-x:auto;overflow-y:auto;width:auto;height:auto;"> <style>#ufybehlmra table { font-family: system-ui, 'Segoe UI', Roboto, Helvetica, Arial, sans-serif, 'Apple Color Emoji', 'Segoe UI Emoji', 'Segoe UI Symbol', 'Noto Color Emoji'; -webkit-font-smoothing: antialiased; -moz-osx-font-smoothing: grayscale; } #ufybehlmra thead, #ufybehlmra tbody, #ufybehlmra tfoot, #ufybehlmra tr, #ufybehlmra td, #ufybehlmra th { border-style: none; } #ufybehlmra p { margin: 0; padding: 0; } #ufybehlmra .gt_table { display: table; border-collapse: collapse; line-height: normal; margin-left: auto; 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} #ufybehlmra .gt_left { text-align: left; } #ufybehlmra .gt_center { text-align: center; } #ufybehlmra .gt_right { text-align: right; font-variant-numeric: tabular-nums; } #ufybehlmra .gt_font_normal { font-weight: normal; } #ufybehlmra .gt_font_bold { font-weight: bold; } #ufybehlmra .gt_font_italic { font-style: italic; } #ufybehlmra .gt_super { font-size: 65%; } #ufybehlmra .gt_footnote_marks { font-size: 75%; vertical-align: 0.4em; position: initial; } #ufybehlmra .gt_asterisk { font-size: 100%; vertical-align: 0; } #ufybehlmra .gt_indent_1 { text-indent: 5px; } #ufybehlmra .gt_indent_2 { text-indent: 10px; } #ufybehlmra .gt_indent_3 { text-indent: 15px; } #ufybehlmra .gt_indent_4 { text-indent: 20px; } #ufybehlmra .gt_indent_5 { text-indent: 25px; } </style> <table class="gt_table" data-quarto-disable-processing="false" data-quarto-bootstrap="false"> <thead> <tr class="gt_heading"> <td colspan="3" class="gt_heading gt_title gt_font_normal gt_bottom_border" style>Comparação entre Juros Contínuos e Aproximação de Taylor</td> </tr> <tr class="gt_col_headings"> <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="Montante Exato">Montante Exato</th> <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="Montante Aproximado (Taylor)">Montante Aproximado (Taylor)</th> <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="Erro">Erro</th> </tr> </thead> <tbody class="gt_table_body"> <tr><td headers="Montante Exato" class="gt_row gt_center">1.051,27</td> <td headers="Montante Aproximado (Taylor)" class="gt_row gt_center">1.051,25</td> <td headers="Erro" class="gt_row gt_center">0,02</td></tr> </tbody> </table> </div> --- ### Funções Marginais em Economia -- A análise marginal é o estudo das taxas de variação das quantidades econômicas. **Exemplo**: Suponha que o **custo total** para a fabricação de `\(x\)` refrigeradores seja dado pela função: `\(C(x) = 8000 + 200x - 0{,}2x^2, \quad \text{com } 0 \leq x \leq 400\)` a) Qual o custo total envolvido na fabricação do 251º refrigerador? -- O custo do 251º refrigerador pode ser estimado pela diferença `\(C(251) - C(250)\)`: `\(= [8000 + 200(251) - 0{,}2(251)^2] - [8000 + 200(250) - 0{,}2(250)^2]\)` -- `\(= 8000 + 50200 - 0{,}2 \cdot 63001 - \left(8000 + 50000 - 0{,}2 \cdot 62500\right)\)` `\(= 45.599,80 - 45500 = 99,80\)` --- ### Funções Marginais em Economia -- b) Determine a taxa de variação da função custo total com relação a `\(x\)` quando `\(x = 250\)`. A taxa de variação instantânea é dada pela derivada da função `\(C(x)\)`: `\(C'(x) = \frac{d}{dx}[8000 + 200x - 0{,}2x^2] = 200 - 0{,}4x\)` -- Substituindo `\(x = 250\)`: `\(C'(250) = 200 - 0{,}4 \cdot 250 = 200 - 100 = 100\)` Ou seja, no ponto `\(x = 250\)`, o custo está crescendo à razão de **100 dólares por refrigerador**. --- ## Função Receita -- Uma função receita `\(R(x)\)` fornece a receita obtida por uma empresa com a venda de `\(x\)` unidades de certo bem: `\(R(x) = p \cdot x\)`. Se o preço unitário de venda `\(p\)` está relacionado com a quantidade de bens `\(x\)`, `\(p = f(x)\)`. Assim, a **função receita** `\(R\)` é dada por: $$ R(x) = p \cdot x = x \cdot f(x) $$ -- A receita marginal é o faturamento conseguido com a venda de uma unidade a mais de um bem. Assim, a função receita marginal como R'(x) é a derivada R' da função R. --- ### Funções Marginais em Economia -- Suponha que a relação entre o preço unitário `\(p\)` e a quantidade demandada `\(x\)` de um produto seja dada pela equação: `\(p = -0{,}02x + 400 \quad \text{com } 0 \leq x \leq 20\,000\)`. a) Determinar a função receita `\(R(x)\)` `\(R(x) = p \cdot x = x(-0{,}02x + 400)=-0{,}02x^2 + 400x\)` -- b) Determinar a função receita marginal `\(R'(x)\)` A derivada da função: `\(R(x)\)` é `\(R'(x) = \frac{d}{dx}(-0{,}02x^2 + 400x) = -0{,}04x + 400\)` --- ### Funções Marginais em Economia -- c) Calcular `\(R'(2.000)\)`. Substituindo na derivada: `\(R'(2\,000) = -0{,}04(2\,000) + 400 = -80 + 400 = 320\)`. **Interpretação:** A receita marginal no ponto `\(x = 2.000\)` é de **R$ 320**, o que indica que a venda do **2.001º** produto gera, aproximadamente, *R$ 320** adicionais de receita. --- ### Diferenciação Implícita e Taxas Relacionadas -- Até o momento, temos lidado com funções na forma y = j{x), ou seja, funções na qual a variável dependente y é expressada explicitamente em termos da variável independente x. Entretanto, nem todas as funções são expressas dessa forma. Em alguns casos, uma função `\(y\)` não é definida explicitamente, ou seja, na forma `\(y = f(x)\)`. -- Por exemplo: na função `\(x^2 + y^2 = 25\)` não é muito simplies isolar `\(y\)` em termos de `\(x\)` de forma explícita. Para encontrar `\(\frac{dy}{dx}\)`, usamos a **derivação implícita**, que consiste em derivar ambos os lados da equação com relação a `\(x\)`, tratando `\(y\)` como uma **função de `\(x\)`**, ou seja, aplicando a **regra da cadeia** sempre que derivamos termos que envolvem `\(y\)`. --- ### Exemplo Diferenciação Implícita -- Considere a equação: `\(x^2 + y^2 = 25\)` Derivando ambos os lados com relação a `\(x\)`: $$ `\begin{aligned} \frac{d}{dx}(x^2) + \frac{d}{dx}(y^2) &= \frac{d}{dx}(25) \quad & \Rightarrow & \quad 2x + \frac{dy}{dx} \cdot 2y = 0 \end{aligned}` $$ -- Isolando `\(\frac{dy}{dx}\)`: $$ \frac{dy}{dx} = \frac{-2x}{2y} = \frac{-x}{y} $$ Assim, a derivada de `\(y\)` em relação a `\(x\)` é `\(\frac{-x}{y}\)`, mesmo sem saber a função `\(y(x)\)` de forma explícita. --- ### Aplicações da Primeira Derivada -- - Teorema: função crescente e função descrescente: Seja `\(f\)` uma função derivável em um intervalo `\((a, b)\)`. Então: -- a. Se `\(f'(x) > 0\)` para cada valor de `\(x\)` em `\((a, b)\)`, então `\(f\)` é **crescente** em `\((a, b)\)`. -- b. Se `\(f'(x) < 0\)` para cada valor de `\(x\)` em `\((a, b)\)`, então `\(f\)` é **decrescente** em `\((a, b)\)`. -- c. Se `\(f'(x) = 0\)` para cada valor de `\(x\)` em `\((a, b)\)`, então `\(f\)` é **constante** em `\((a, b)\)`. --- ### Extremos Relativos -- A função `\(f\)` tem um **máximo relativo** em `\(x = c\)` se existe um intervalo aberto `\((a, b)\)` contendo `\(c\)`, tal que `\(f(x) \leq f(c) \quad \text{para todos os } x \in (a, b)\)` -- Uma função `\(f\)` tem um **mínimo relativo** em `\(x = c\)` se existe um intervalo aberto `\((a, b)\)` contendo `\(c\)`, tal que `\(f(x) \geq f(c) \quad \text{para todos os } x \in (a, b)\)` --- ### Extremos Relativos -- <img src="data:image/png;base64,#index_files/figure-html/unnamed-chunk-16-1.png" style="display: block; margin: auto;" /> --- ### Teste da Segunda Derivada -- Seja `\(f\)` uma função derivável duas vezes em um ponto `\(c\)`: - `\(f'(c) = 0\)` e `\(f''(c) > 0\)`, então `\(f\)` tem **mínimo relativo** em `\(c\)`. - `\(f'(c) = 0\)` e `\(f''(c) < 0\)`, então `\(f\)` tem **máximo relativo** em `\(c\)`. - `\(f'(c) = 0\)` e `\(f''(c) = 0\)`, o teste é **inconclusivo**. -- - **Concavidade de uma Função** - Se `\(f''(x) > 0\)` para todo `\(x\)` em um intervalo, então `\(f\)` é **côncava para cima** nesse intervalo. - Se `\(f''(x) < 0\)`, então `\(f\)` é **côncava para baixo**. - Se `\(''(x) = 0\)` e muda de sinal, há um **ponto de inflexão**. --- ### Aplicação Segunda Derivada -- <img src="data:image/png;base64,#index_files/figure-html/unnamed-chunk-17-1.png" style="display: block; margin: auto;" /> --- # Integração -- - Se conhecemos ataxa devariação de uma quantidade em relação a outra, podemos determinar a relação entre as duas quantidades? - Uma função `\(F\)` é uma **antiderivada** de `\(f\)` em um intervalo `\(J\)` se: `\(F'(x) = f(x) \quad \text{para todo } x \in J.\)` -- Isso significa que a derivada de `\(F\)` é igual à função original `\(f\)`. Ou seja, encontrar uma antiderivada é o processo inverso da diferenciação. > Exemplo: Se `\(f(x) = 2x\)`, então uma antiderivada é `\(F(x) = x^2\)`, pois, `\(\frac{d}{dx}(x^2) = 2x = f(x)\)`. --- # Integração -- Outro exemplo: - Seja `\(f(x) = 3x^2 + 2\)`, então `\(F(x) = x^3 + 2x + 1\)` é uma antiderivada de `\(f\)`, pois `\(F'(x) = \frac{d}{dx}(x^3 + 2x + 1) = 3x^2 + 2 = f(x)\)` - Uma função pode ter infinitas antiderivadas, que diferem apenas por uma constante. Por exemplo, todas as funções da forma: `\(F(x) = x^2 + C\)` são antiderivadas de `\(f(x) = 2x\)`, para qualquer constante real `\(C\)`. --- # Integração -- - O processo de determinar todas as antiderivadas de uma função é chamado **antidiferenciação**, ou **integração**. -- - Usamos o símbolo `\(\int\)`, denominado **sinal da integral**, para indicar que a operação de integração deve ser executada sobre uma função `\(f\)`. Assim: $$ \int f(x)\, dx = F(x) + C $$ Leia-se: *a integral indefinida de `\(f(x) \, dx\)` em relação a `\(x\)` é igual a `\(F(x)\)` mais `\(C\)`*". -- Isso nos diz que a **integral indefinida** de `\(f\)` é a **família de funções** dada por `\(F(x) + C\)`, onde `\(F'(x) = f(x)\)` --- # Regras de Integração -- - **A Integral Indefinida de uma Constante** Se `\(k\)` é uma constante, então: `\(\int k \, dx = kx + C\)` -- - **A Integral de uma Potência de `\(x\)`** Se `\(n \neq -1\)`, então: `\(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\)` -- - **Fator Constante** Se `\(c\)` é uma constante, então: `\(\int c \cdot f(x) \, dx = c \int f(x) \, dx\)` --- # Regras de Integração -- - **Integral de uma Soma ou Diferença** $$ \int [f(x) \pm g(x)] \, dx = \int f(x) \, dx \pm \int g(x) \, dx $$ -- - **A Integral da Função Exponencial** `\(\int e^x \, dx = e^x + C\)` Mais geralmente, se temos a função `\(e^{ax}\)`, com `\(a \neq 0\)`, então: `\(\int e^{ax} \, dx = \frac{1}{a} e^{ax} + C\)` --- ## Exemplos com as Propriedades Vamos aplicar algumas propriedades básicas da integração indefinida para resolver integrais simples. a. `\(\int 5 \, dx = 5x + C\)` -- b. `\(\int (3x^2 + 4x) \, dx = \int 3x^2 \, dx + \int 4x \, dx = x^3 + 2x^2 + C\)` -- c. `\(\int 7x^3 \, dx = 7 \cdot \int x^3 \, dx = 7 \cdot \frac{x^4}{4} + C = \frac{7}{4}x^4 + C\)` -- d. `\(\int e^x \, dx = e^x + C\)` -- e. `\(\int x^4 \, dx = \frac{x^5}{5} + C\)` --- # Integral Definida -- A **integral definida** de uma função contínua `\(f(x)\)` no intervalo$[a, b]$ representa a **área sob a curva** de `\(f\)` entre `\(x = a\)` e `\(x = b\)`, considerando as áreas abaixo do eixo `\(x\)` como negativas. -- Formalmente, a **integral definida** é o limite de uma soma de Riemann: `$$\int_a^b f(x)\, dx = \lim_{n \to \infty} \sum_{i=1}^n f(x_i^*) \Delta x$$` -- onde: `\([a, b]\)` é dividido em `\(n\)` subintervalos de comprimento `\(\Delta x = \frac{b-a}{n}\)`; `\(x_i^*\)` é um ponto arbitrário dentro do `\(i\)`-ésimo subintervalo; `\(f(x_i^*) \Delta x\)` é a área aproximada de um retângulo sob a curva. --- # Interpretação Gráfica <img src="data:image/png;base64,#img/Integral_riemann_animada.gif" width="40%" style="display: block; margin: auto;" /> <div style="text-align: center; font-size: 80%;"> Fonte: <a href="https://pt.m.wikipedia.org/wiki/Integral">Wikipedia</a> </div> --- # O Teorema Fundamental do Cálculo -- Seja `\(f\)` contínua no intervalo `\([a, b]\)`. Então `$$\int_a^b f(x)\, dx = F(b) - F(a)$$` onde `\(F\)` é uma **antiderivada qualquer** de `\(f\)`, ou seja, `\(F'(x) = f(x)\)`. -- Ao aplicar o **Teorema Fundamental do Cálculo**, é conveniente usar a notação `$$\int_a^b f(x)\,dx = F(x)\Big|_a^b = F(b) - F(a)$$` --- ## Propriedades da Integral -- Sejam `\(f\)` e `\(g\)` funções contínuas em um intervalo que contenha os pontos envolvidos, então: 1. `\(\int_a^a f(x)\,dx = 0\)` -- 2. `\(\int_b^a f(x)\,dx = -\int_a^b f(x)\,dx\)` -- 3. Se `\(c\)` é uma constante, então: `\(\int_a^b c f(x)\,dx = c \int_a^b f(x)\,dx\)` -- 4. A integral da soma é a soma das integrais: `\(\int_a^b [f(x) + g(x)]\,dx = \int_a^b f(x)\,dx + \int_a^b g(x)\,dx\)` -- 5. Se `\(a < c < b\)`, então: `\(\int_a^b f(x)\,dx = \int_a^c f(x)\,dx + \int_c^b f(x)\,dx\)` --- ## Integração por Substituição -- Considere a integral indefinida: `\(\int 2(2x + 4)^5 \, dx\)` -- 🔁 Etapa 1 - Substituição: Seja `\(u = 2x + 4 \quad \Rightarrow \quad du = 2 \, dx\)`. Então, isolando `\(dx\)`: `\(dx = \frac{du}{2}\)` e substituindo na integral (1): `\(\int 2(2x + 4)^5 \, dx = \int (2x + 4)^5 \cdot 2 \, dx = \int u^5 \cdot du\)` -- 🔁 Etapa 2 - Resolvando a nova Integral usando a *Regra da Potência**: `\(\int u^5 \, du = \frac{u^6}{6} + C\)`. -- 🔁 Etapa 3: Retornar à variável original: Como `\(u = 2x + 4\)`, substituímos de volta: `\(\int 2(2x + 4)^5 \, dx = \frac{(2x + 4)^6}{6} + C\)`. -- 🔍 Verificação: `\(\frac{d}{dx} \left( \frac{(2x + 4)^6}{6} \right)\)` = `\(\frac{1}{6} \cdot 6(2x + 4)^5 \cdot 2 = 2(2x + 4)^5\)` --- ## Funções de Várias Variáveis -- Uma **função real de duas variáveis** $ f$ consiste em: 1. Um conjunto `\(A\)` de **pares ordenados** de números reais `\((x, y)\)`, chamado **domínio** da função. 2. Uma **regra de associação** que atribui a cada par ordenado `\((x, y) \in A\)` um único número real, denotado por `\(z = f(x, y)\)`. -- 🌐 Exemplo: A função `\(f(x, y) = x^2 + y^2\)` é uma função de duas variáveis cujo valor em cada ponto do plano é dado pela soma dos quadrados das coordenadas. O domínio dessa função é: `\(D = \mathbb{R}^2 = \{ (x, y) \in \mathbb{R} \times \mathbb{R} \}\)` e para cada par `\((x, y)\)`, temos: `\(f(x, y) = x^2 + y^2\)` --- ### 📊 Visualização Gráfica <div class="plotly html-widget html-fill-item-overflow-hidden html-fill-item" id="htmlwidget-f85a982514ce28ae593b" style="width:504px;height:504px;"></div> <script type="application/json" data-for="htmlwidget-f85a982514ce28ae593b">{"x":{"visdat":{"28a4479a25e":["function () ","plotlyVisDat"]},"cur_data":"28a4479a25e","attrs":{"28a4479a25e":{"x":{},"y":{},"z":{},"alpha_stroke":1,"sizes":[10,100],"spans":[1,20],"type":"surface","inherit":true}},"layout":{"margin":{"b":40,"l":60,"t":25,"r":10},"title":"Gráfico da função f(x, y) = x² + y²","scene":{"xaxis":{"title":"x"},"yaxis":{"title":"y"},"zaxis":{"title":"f(x, 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--- ## Derivadas Parciais -- Suponha que `\(f(x, y)\)` seja uma função de duas variáveis `\(x\)` e `\(y\)`. Então, a **derivada parcial de primeira ordem de `\(f\)`** com relação a `\(x\)`, no ponto `\((x, y)\)`, é dada por: $$ \frac{\partial f}{\partial x} = \lim_{h \to 0} \frac{f(x + h, y) - f(x, y)}{h} $$ Desde que esse limite exista. -- Analogamente, a **derivada parcial de primeira ordem de `\(f\)`** com relação a `\(y\)`, no ponto `\((x, y)\)`, é: $$ \frac{\partial f}{\partial y} = \lim_{k \to 0} \frac{f(x, y + k) - f(x, y)}{k} $$ --- ## Exemplo -- Considere a função `\(f(x, y) = x^2 - x y^2 + y^3\)`. A Derivada parcial em relação a `\(x\)` é `\(\frac{\partial f}{\partial x} = 2x - y^2\)`. Já a Derivada parcial em relação a `\(y\)` é `\(\frac{\partial f}{\partial y} = -2xy + 3y^2\)` -- Taxas de variação no ponto `\((1, 2)\)`: Na direção `\(x\)`: `\(\frac{\partial f}{\partial x}(1, 2) = 2(1) - (2)^2 = -2\)` > A função **diminui 2 unidades** para cada aumento de 1 unidade em \( x \), com \( y \) constante. -- Taxas de variação no ponto `\((1, 2)\)`: Na direção `\(y\)`: `\(\frac{\partial f}{\partial y}(1, 2) = -2(1)(2) + 3(2)^2 = -4 + 12 = 8\)` > A função **aumenta 8 unidades** para cada aumento de 1 unidade em \( y \), com \( x \) constante. --- ### Função de Produção de Cobb-Douglas -- Considere a função de produção: `\(f(x, y) = a x^b y^{1-b}\)`. Com `\(a\)` e `\(b\)` são constantes positivas com `\(0 < b < 1\)`; `\(x\)` representa o valor gasto com **mão de obra**; `\(y\)` representa o valor gasto com **capital** (equipamentos, prédios etc.) e `\(f(x, y)\)` representa a **produção total**. -- - **Produtividade Marginal da Mão de Obra**: `\(\frac{\partial f}{\partial x} = ab x^{b-1} y^{1-b}\)` > Mede a **taxa de variação da produção** com relação ao gasto em mão de obra, mantendo o capital constante. -- - **Produtividade Marginal do Capital**: `\(\frac{\partial f}{\partial y} = a(1 - b) x^b y^{-b}\)` > Mede a **taxa de variação da produção** com relação ao gasto em capital, mantendo constante o gasto em mão de obra. --- ### Função de Produção de Cobb-Douglas -- <div class="plotly html-widget html-fill-item-overflow-hidden html-fill-item" id="htmlwidget-de0f735d05eac7555526" style="width:504px;height:504px;"></div> <script type="application/json" 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--- ### Exemplo: Função Cobb-Douglas -- A produção de certo país é descrita pela função: `\(f(x, y) = 30x^{2/3}y^{1/3}\)`, com `\(x\)` representando unidades de mão de obra e `\(y\)` unidades de capital. a) Calculer `\(f_x\)` e `\(f_y\)` - Derivada parcial em relação a `\(x\)`: `\(f_x = \frac{\partial}{\partial x}(30x^{2/3}y^{1/3}) = 30 \cdot \frac{2}{3} x^{-1/3} y^{1/3} = 20x^{-1/3}y^{1/3}\)` -- - Derivada parcial em relação a `\(y\)`: `\(f_y = \frac{\partial}{\partial y}(30x^{2/3}y^{1/3}) = 30x^{2/3} \cdot \frac{1}{3} y^{-2/3} = 10x^{2/3}y^{-2/3}\)` --- ### Exemplo: Função Cobb-Douglas -- b) Produtividade marginal no ponto `\((x = 125, y = 27)\)` - Produtividade marginal de mão de obra: `\(f_x(125, 27) = 20 \cdot 125^{-1/3} \cdot 27^{1/3} = 20 \cdot \frac{1}{5} \cdot 3 = 12\)` - Produtividade marginal de capital: `\(f_y(125, 27) = 10 \cdot 125^{2/3} \cdot 27^{-2/3} = 10 \cdot 25 \cdot \frac{1}{9} = \frac{250}{9} \approx 27.78\)` -- c) Interpretação Econômica - O aumento de uma unidade em **capital** gera aproximadamente **27.78 unidades** adicionais de produção. Já o aumento de uma unidade em **mão de obra** gera apenas **12 unidades** adicionais de produção. **Conclusão:** O governo deveria ter incentivado **investimentos em capital**, pois proporcionam um maior aumento de produtividade em comparação à mão de obra nesse contexto. --- ### Função de Produção de Cobb-Douglas -- <div class="plotly html-widget html-fill-item-overflow-hidden html-fill-item" id="htmlwidget-9eefd07531f021030c44" style="width:504px;height:504px;"></div> <script type="application/json" data-for="htmlwidget-9eefd07531f021030c44">{"x":{"visdat":{"28a426782d77":["function () 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--- # Referências -- .pull-left[ <img src="data:image/png;base64,#img/livroStTan.jpg" width="40%" style="display: block; margin: auto;" /> <div style="text-align: center; font-size: 80%;"> <a href="https://www.amazon.com.br/Matem%C3%A1tica-aplicada-administra%C3%A7%C3%A3o-economia-Tan/dp/8522116466">S. T. TAN</a> </div> ] .pull-right[ <img src="data:image/png;base64,#img/Livro2.jpg" width="40%" style="display: block; margin: auto;" /> <div style="text-align: center; font-size: 80%;"> <a href="https://shopee.com.br/Livro-Matem%C3%A1tica-Aplicada-%C3%80-Administra%C3%A7%C3%A3o-Economia-e-Contabilidade-de-Afr%C3%A2nio-Murolo-Gi%C3%A1como-Bonetto-i.403437664.19898010034"> Afrânio Murolo & Giácomo Bonetto</a> </div> ]